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Parametric curves 3 Usually describe curve in plane with equation in x and y, e.g., x 2 + y 2 = r 2 is circle at origin In other words, can write y as a function of x or vice-versa. Problems Form may not be easy to work with Lots of curves that can't write as single equation in only x and y

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Parametric curves 4 One solution is to use a parametric equation. In it, define both x and y to be functions of a third variable, say t x = f(t) y = g(t) Each value of t defines a point (x,y)=( f(t), g(t) ) that we can plot. Collection of points we get by letting t take on all its values is a parametric curve

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Phase plane plot 23 If have initial condition (other than previous 3) that is exactly on curve (red dot) can tell its path in phase plane. Q: What if not on curve but very close to it (yellow dot)? A: ?

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Phase plane plot 24 To help understand solution for any initial condition, can make phase plot and add information about how each state variable changes with time, i.e., display the first derivative of each state variable.

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Phase plane plot 25 Will show rate of change of state variables at a point by drawing a vector point there. Horizontal component of vector is rate of change of variable one; vertical component of vector is rate of change of variable two. y 1 (t) y 2 (t) y ' 1 (t) y ' 2 (t)

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Phase plane plot 27 To plot vectors at point, use quiver( x, y, u, v ) This plots the vectors (u,v) at every point (x,y) x is matrix of x-values of points y is matrix of y-values of points u is matrix of horizontal components of vectors v is matrix of vertical components of vectors All matrices must be same size v (x,y) u

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Phase plane plot 33 Try It To see solution path for specific initial conditions, imagine dropping a toy boat (initial condition) at a spot in a river (above plot) and watching how current (arrows) pushes it around.

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Phase plane plot 35 From phase-plane plot it appears reasonable to say that if the initial conditions of the solutions of a differential equation are close to each other, the solutions are also close to each other.

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Chaos 50 Chaotic systems are: Deterministic – no randomness involved – If start with identical initial conditions, get identical final states High sensitivity to initial conditions – Tiny differences in starting state can lead to enormous differences in final state, even over small time ranges Seemingly random – Unexpected and abrupt changes in state occur Often sensitive to slight parameter changes

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Chaos 51 In 1963, Edward Lorenz, mathematician and meteorologist, published set of equations Simplified model of convection rolls in the atmosphere Also used as simple model of laser and dynamo (electric generator)

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Chaos 52 Set of equations Nonlinear Three-dimensional Deterministic, i.e., no randomness involved Important implications for climate and weather prediction – Atmospheres may exhibit quasi-periodic behavior and may have abrupt and seemingly random change, even if fully deterministic – Weather can't be predicted too far into future!

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Chaos 53 Equations, in state-space form, are * Notice only two terms have nonlinearities * Also appear in slightly different forms

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Chaos 68 Lorenz attractor shows us some characteristics of chaotic systems Paths in phase space can be very complicated Paths can have abrupt changes at seemingly random times Small variations in a parameter can produce large changes in trajectories

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Chaos 79 So even though initial conditions only differ by 1 / 100,000 of a percent, the trajectories become very different!

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Chaos 80 This extreme sensitivity to initial conditions is often called The Butterfly Effect A butterfly flapping its wings in Brazil can cause a tornado in Texas.

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Chaos 81 Lorenz equations are good example of chaotic system Deterministic High sensitivity to initial conditions – Very tiny differences in starting state can lead to substantial differences in final state Unexpected and abrupt changes in state Sensitive to slight parameter changes

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Chaos 82 MATLAB is good for studying chaotic systems Easy to set and change initial conditions or parameters Solving equations is fast and easy Plotting and comparing 2D and 3D trajectories also fast and easy